The course provides introductory material on general relativity and its mathematical language of differential geometry. This is followed by more advanced modules with applications to the study of black holes, cosmology and aspects of general relativity related to string theory. There is a year-long introduction to quantum field theory which introduces the famous Feynman diagrams of particle physics in a systematic way, and studies aspects of modern particle physics. There is also an introduction to the concepts of quantum information theory.
The maths course assumes students have a familiarity with quantum mechanics and special relativity at an introductory level. No prior knowledge of general relativity is assumed.
Taken full-time, the course lasts a full year, starting in one September and ending during the following September.
The course has a simple structure, consisting of 180 credits, split into 120 credits of taught modules during the autumn and spring semesters and a 60-credit research project that is completed in the summer period. All modules are compulsory.
Modules are mainly delivered through lectures and example and/or problem classes. You will typically be assessed by an examination at the end of the semester in which a given module is taught. However a small proportion of the assessment is by coursework, essay and student presentation.
During the summer period, you will concentrate on an independent maths research project under the supervision of a member of academic staff, writing a substantial dissertation.
Here are a few maths books that can be used to brush up on prerequisite material.
- Classical Mechanics - Classical Mechanics by Kibble and Berkshire (In particular, it is worth looking at Lagrangian and Hamiltonian mechanics in Ch10, 11 and 12).
- Quantum Mechanics - Quantum Mechanics Demystified by David McMahon
- Special Relativity - Flat and curved space-times by George F. Ellis and Ruth M. Williams (The first part covers special relativity and the second part will be useful for the MSc course).
- Mathematical Methods - Mathematical Methods for Physics and Engineering: A Comprehensive Guide by K. F. Riley, M. P. Hobson, S. J. Bence (the key topics are vector calculus/analysis, Fourier series and Fourier transforms, the Laplace equation, the heat equation, the wave equation, complex variables and contour integration).
For the modules taught during the MSc, here are a few suggestions for preliminary reading that will introduce some of the ideas in a fairly non-technical way. These aren't supposed to cover all the module material but are there just to get you started.
- General Relativity, black holes and cosmology: Flat and curved space-times by George F. Ellis and Ruth M. Williams.
- Gravity: An Introduction to Einstein's General Relativity by J.B. Hartle.
- Quantum Field Theory: Quantum Field Theory in a Nutshell by A. Zee. This book has a very good discussion of the concepts but does get to quite advanced topics (some of which are not in the MSc).
- QED The Strange Theory of Light and Matter, by R.P. Feynman.
- Differential Geometry
The modern study of general relativity and particle theory requires familiarity with a number of tools of differential geometry. These are introduced in this module and include manifolds, tensors, symmetries, Lie Groups, connections and differentiation and integration on manifolds.
- Quantum Information Science
The paradigm of Quantum Information Science (QIS) is that quantum devices made of systems such as atoms and photons, can outperform the present day technology in key applications ranging from computing power and communication security to precision measurements. Quantum information processing and the measurement and control of individual quantum systems are central topics in QIS, lying at the intersection of quantum mechanics with "classical" disciplines such as information theory, probability and statistics, computer science and control engineering.
After a short review of the necessary probabilistic notions, the first part introduces the operational framework of quantum theory involving the fundamental concepts of states, measurements, quantum channels, instruments. This includes some of the influential results in the field such as entanglement and quantum teleportation, Bell's theorem and the quantum no-cloning theorem. The second part covers at least two topics.
The module is an introduction to general relativity and the observed gravitational phenomena that are explained by the theory. Topics include: elementary geometry, a revision of special relativity, the physical basis of general relativity, the metric and Einstein's equation, the Schwarzschild solution and observational tests of general relativity, gravitational collapse to a black hole.
- Quantum Field Theory
Quantum Field Theory is the study of the quantum dynamics of relativistic particles. The module gives the quantum description of the electrons, photons and other elementary particles, including a discussion of spin, and bosons and fermions. Lectures will provide an introduction to functional integrals, Feynman diagrams, and the standard model of particle physics.
- Advanced Gravity
In this course we will develop the ideas behind General Relativity (GR) to an advanced level. As we will explain, GR is based on the geometry of four dimensional spacetime, the curvature of which is governed by the Einstein’s equations. Some solutions to these equations will be presented, including black holes and cosmological solutions. Gravity in the weak field limit will be derived from the full theory, demonstrating how one should understand the gravitational interaction in terms of graviton exchange. The course will then move on to advanced topics. This includes modified gravity models (eg models with extra dimensions) that are at the forefront of current research.
- Black Holes
General relativity predicts the existence of black holes which are regions of space-time into which objects can be sent but from which no classical objects can escape. This module uses techniques learnt in Differential Geometry to systematically study black holes and their properties, including horizons and singularities. Astrophysical processes involving black holes are discussed, and there is a brief introduction to black hole radiation discovered by Hawking.
- Modern Cosmology
This module introduces students to the key ideas behind modern approaches to our understanding of the role of inflation in the early and late universe, in particular through the formation of structure, the generation of anisotropies in the cosmic microwave background radiation, and the origin of dark energy.
Topics include: a brief review of Friedmann models and hot big bang, inflation and why it is required, fluctuations from inflation, structure formation, gravitational lensing: what it is, and using it to detect dark matter, cosmic microwave background anisotropies, dark energy, the cosmological constant, extra dimensions, modified gravity, and the string landscape.
- Gravity, Particles and Fields Dissertation
In this module a substantial investigation will be carried out on a topic related to the taught modules of the course. The study will be largely self-directed, with oversight and input provided where necessary by a supervisor from the School of Mathematical
Sciences or the School of Physics and Astronomy. The topic will be chosen by agreement between the student and supervisor. The topic could be based on a theoretical investigation, a literature review, or a combination of the two.
Students successfully completing the course should have demonstrated:
- Knowledge and understanding of a range of mathematical core concepts and results in gravitation and quantum theory.
- Knowledge and understanding of some advanced concepts and techniques related to current research in gravitation and quantum theory.
- Awareness of some current problems and new insights in gravitation and quantum theory.
- Conceptual understanding that enables the critical evaluation of current research, methodology or advanced scholarship.
- The ability to apply knowledge in the discipline to novel problems.
Students successfully completing the course should be able to:
- Apply complex concepts, methods and techniques to familiar and novel situations.
- Work with abstract concepts and in a context of generality.
- Reason logically and work analytically.
- Perform with high level of accuracy.
- Relate mathematical results to their physical applications.
- Transfer expertise between different topics in mathematical physics.
- Select and apply appropriate methods and techniques to solve problems.
- Justify conclusions using mathematical arguments with appropriate rigour.
- Communicate results using appropriate styles, conventions and terminology.
- Use appropriate IT packages effectively.
- Communicate with clarity.
- Work effectively, independently and under direction.
- Adopt effective strategies for study.